A convexity criterion via the De Giorgi slope

Let $X$ be a Banach space and $f\in {\mathcal{C}}^1(X)$ be bounded from below. We show that if for some $m\ge 1$, the function $x\mapsto \mathrm{\parallel }\mathrm{\nabla }f(x){\mathrm{\parallel }}^m$ is convex, then $f$ is convex. We also establish a more general version of this result: if $f$ is continuous and bounded from below, then it is convex, provided $x\mapsto s_f(x{)}^m$ is convex for some $m\ge 1$, where $s_f$ denotes the (De Giorgi) metric slope of $f$.